Harmonic superspace: key to N=2 supersymmetry theories

The concept of a harmonic N=2 superspace with additional coordinates related to the SU(2)/U(1) sphere is introduced. This concept leads to an adequate geometric description of the N=2 theories of matter, the Yang-Mills theory, and supergravity in terms of superfields without coupling. A new effect has been discovered: the unboundedness of the number of gauge and auxiliary degrees of freedom.